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A function g(x)g(x) increases by 44 over every unit interval in xx and g(0)=0g(0) = 0. Which could be a function rule for g(x)g(x)?\newlineChoices:\newline(A) g(x)=−4xg(x) = -4x\newline(B) g(x)=4xg(x) = 4^x\newline(C) g(x)=4xg(x) = 4x\newline(D) g(x)g(x) = −x-x - 44

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Q. A function g(x)g(x) increases by 44 over every unit interval in xx and g(0)=0g(0) = 0. Which could be a function rule for g(x)g(x)?\newlineChoices:\newline(A) g(x)=−4xg(x) = -4x\newline(B) g(x)=4xg(x) = 4^x\newline(C) g(x)=4xg(x) = 4x\newline(D) g(x)g(x) = −x-x - 44
  1. Start at origin: Since g(0)=0g(0) = 0, the function starts at the origin. This eliminates any function with a non-zero yy-intercept.
  2. Determine linear function: The function increases by 44 for each increase of 11 in xx, so the slope must be 44. This means the function is linear and of the form g(x)=4x+bg(x) = 4x + b.
  3. Substitute x=0x=0: Since g(0)=0g(0) = 0, we can substitute x=0x = 0 into the equation to find bb: g(0)=4(0)+bg(0) = 4(0) + b, which simplifies to 0=b0 = b.
  4. Find function rule: Now we know that b=0b = 0, so the function rule is g(x)=4xg(x) = 4x.
  5. Match function rule: Looking at the choices, (C) g(x)=4xg(x) = 4x matches our function rule.

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